Mathematical Analysis for Quadrilateral Rotated $Q_1$ Element II: Poincarè Inequality and Trace Inequality

Abstract

This is the second part of the paper for the mathematical study of nonconforming rotated $Q_1$ element (NR$Q_1$ hereafter) on arbitrary quadrilateral meshes. Some Poincarè Inequalities are proved without assuming the quasi-uniformity of the mesh subdivision. A discrete trace inequality is also proved.